Properties

Label 2-9072-1.1-c1-0-24
Degree $2$
Conductor $9072$
Sign $1$
Analytic cond. $72.4402$
Root an. cond. $8.51118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.267·5-s + 7-s + 6.19·11-s − 6.46·13-s − 7·17-s − 0.732·19-s − 4.19·23-s − 4.92·25-s − 1.53·29-s − 8.19·31-s − 0.267·35-s + 10.6·37-s − 2.53·41-s + 1.46·43-s + 4.73·47-s + 49-s + 9.46·53-s − 1.66·55-s + 4.19·59-s + 3.92·61-s + 1.73·65-s + 6.73·67-s + 6.53·71-s + 8.26·73-s + 6.19·77-s + 9.12·79-s + 16.5·83-s + ⋯
L(s)  = 1  − 0.119·5-s + 0.377·7-s + 1.86·11-s − 1.79·13-s − 1.69·17-s − 0.167·19-s − 0.874·23-s − 0.985·25-s − 0.285·29-s − 1.47·31-s − 0.0452·35-s + 1.75·37-s − 0.396·41-s + 0.223·43-s + 0.690·47-s + 0.142·49-s + 1.29·53-s − 0.223·55-s + 0.546·59-s + 0.502·61-s + 0.214·65-s + 0.822·67-s + 0.775·71-s + 0.967·73-s + 0.706·77-s + 1.02·79-s + 1.82·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9072\)    =    \(2^{4} \cdot 3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(72.4402\)
Root analytic conductor: \(8.51118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.648384105\)
\(L(\frac12)\) \(\approx\) \(1.648384105\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 0.267T + 5T^{2} \)
11 \( 1 - 6.19T + 11T^{2} \)
13 \( 1 + 6.46T + 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 + 4.19T + 23T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
31 \( 1 + 8.19T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 1.46T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 9.46T + 53T^{2} \)
59 \( 1 - 4.19T + 59T^{2} \)
61 \( 1 - 3.92T + 61T^{2} \)
67 \( 1 - 6.73T + 67T^{2} \)
71 \( 1 - 6.53T + 71T^{2} \)
73 \( 1 - 8.26T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.64203555595609987731472331670, −7.03181860490831556903116049342, −6.48357935773604818680963619290, −5.71151105207996987462752611948, −4.84234298560797510584726947305, −4.14017079774747966870773150680, −3.75458468968097826075802367168, −2.26762568290556402627350848754, −2.02253417661674526010801469183, −0.59895548263554099142157979052, 0.59895548263554099142157979052, 2.02253417661674526010801469183, 2.26762568290556402627350848754, 3.75458468968097826075802367168, 4.14017079774747966870773150680, 4.84234298560797510584726947305, 5.71151105207996987462752611948, 6.48357935773604818680963619290, 7.03181860490831556903116049342, 7.64203555595609987731472331670

Graph of the $Z$-function along the critical line